ROTATION. 



the wheel in order to raife a weight q, which is applied to 

 the circumference of the axle, it is required to affign the 

 quantity q, when the momentum generated in it in any given 

 time fhall be the greateft poflible ; the inertia of the wheel 

 and axle not being confidered. 



Making, as before, S D = n, S A = m, the force which 



. . . nmp — tn- q 



accelerates q is/= — -!■ — =j 



n z p + tri q 



therefore, if / = 193 

 inches, the velocity generated in q in the time t will be 



Ztlx 



ip — 



n'p -+- ni' q 

 be 2t I X 



-, and the momentum generated in q will 



reduced gives q — 



and as this is to be a maximum 

 n '/> + m' q 



by the problem, its fluxion = o, which being taken and 



J («* />' + n>mp) - n r p 



m 1 



Therefore, if S D : S A :: n : I, then q — p ^ (n" + n 3 ) 

 — n z p ; if the radius of the axle equal the radius of the 

 wheel, that is, n — 1 ; then the weight y = /> ( v ' 2 — 1), 

 and confequently the weight moved muft be about T ' T ths of 

 the moving force. 



But if we introduce the weight of the wheel, and call it tv, 

 r being the diftance of the centre of gyration from the axis, 

 then the momentum generated in the time t will be exprefled 



by 2 It x — — — , which is the greateft poflible 



7 <tvr' + n-p + mq 6 



y (n *p > + 2 «"" r'p iv + r % iv z + r* n mp w + 

 n" mp 1 ) — (n'p + r' <w) 



when q = ■ 



Prop. V. 



Again, let A B C H (Jig. 7.) be a fyftem of bodies move- 

 able round a vertical axis, which pafles through the common 

 centre of gravity of the fyftem. And fuppofe D E G to 

 be a wheel, the axis of which is vertical, and coinciding with 

 that of the fyftem ; let motion be communicated by means 

 of a line going round this wheel, the Itring D P being 

 ftretched by a given weight p ; let it be required to aflign 

 the radius of the wheel E G D, fo that the angular velocity 

 communicated to the fyftem in a given time may be the 

 greateft poflible. Let the weight of the fyftem = to, and 

 the diftance of the centre of gyration from the axis of motion 

 = r, the radius fought S D = x ; then the motive force 

 being p, the velocity generated in a given time in that 



— , and 



defcending weight will be proportional to j 



the angular velocity generated in the fame time as 



which is to be a maximum by the conditions of the problem, 



P* 



we have, therefore, 



pnur 1 x + f x'x — 2 j>* 



= O, 



(wr* + p* i y 



/ — , the diftance fought. 

 Suppofing, therefore, the moving force = -J of the weight 



whence/ wr 1 = p 1 *% or * = r 



order to produce the greateft angular velocity in a given 

 time. 



Prop. VI. 



In order to increafe the aftion of a given moving force 

 againft a weight to be raifed, or refiftance to be overcome, a 

 combination of two or more mechanic powers is frequently 

 made ufe of. Thus, let p be a power applied by means of a 

 line to the vertical wheel C, (Jig. 6. ), and fuppole the 

 circumference of the axle K to be in contaft with the cir- 

 cumference of the wheel B, fo that the circumference of 

 the wheel B may always move equally fait with that of the 

 axis which belongs to C ; let alfo the axle of B communi- 

 cate motion to the vertical wheel A, to the axle of which a 

 weight, q, is fufpended, fo as to aft in oppofition to p ; 

 moreover, let Imn to I, be the fum of the ratios of the 

 radius of each wheel to that of its axle; then if plmn 

 = q, the two weights, p and q, will fuftain each other in equi- 

 librio; but if p Im n be greater than q, the equilibrium will 

 be deftroyed, and the weight q will afcend ; and it is re- 

 quired to aflign the fpace which, under thofe circumftances, 

 will be defcribed by q in a given time. 



Let the radii of the wheel and axle A be in the ratio of 

 /to 1; thofe of B as m to I ; and thofe of C as n to 1 ; 

 the diftance of the centre of gyration in A from the axis 

 = r, the fame of B = r\ and of C = r" ; the weight 

 of the wheel and axle A = <w ; that of B = vf ; and that 

 of C = w". 



Now the abfolute moving force is p, but fince q afts in 

 oppofition to it, it muft ba fubdufted from p, in order to 

 obtain the real motive force of the fyftem. And fince q 



would balance a weight = , , if applied at p, the force 

 Imn 



plmn — q 



which impels p on the whole will be ' 



In the 



next place, the inertia which refifts the communication of mo- 

 tion to />'muft be afcertained. Now motion is communicated 

 to the wheel A, from the circumference of the axle B, and 

 the inertia of A, and of the weight q, which refills the com- 

 munication of a force applied at S = j t ; in regard, 



therefore, to the inertia of A and q, thefe may be fuppofed 



f W f A- Q 



to be removed, and the equivalent mafs collefted into 



the circumference of the wheel A, or of the axle B. 

 And fince motion is communicated to B, by the circum- 

 ference of the axle C, the inertia of B, together with 



iv t^ -f- q ... . 

 the equivalent mafs ^ will be 



r> ' V- 



vi r" + q 



I m> 



In like manner, fince motion is communicated to C by the 

 weight p afting at D, the inertia which refills the commu- 

 nication of motion to D or p, will be 



/ n- + -a/" r m r n P V + wr* + q _ 

 — ^ _ _ + T m'n 1 



I' m- n" p + I- m 1 w 



" r"* + r'~ I' to' + v> r- + q 



/ tu . 



•f the fyftem •», we (hall have x = r */ -j— = 2 r ; that is, and the force wn j cn accelerates p in its defceat from reft 



I m n (plmn — q) 



the weight fhould be applied at a diftance from the axis, 

 equal to twice the diftance of the centre of gyration, in 



/' m 1 n 1 / + /' m L -w" r"' + r" 1' w n + t» r 1 + 5' 



and 



